How can I use the covariant derivative to derive the Riemann curvature tensor?

[e-pie]

Homework Statement

If $$A_i$$ is a covariant vector such that $$A_{i,j}+A_{j,i}=0$$, show that $$A_{i,jk}=-A_rR^r_{kij}$$ where $$R^r_{kij}$$ is the Riemann curvature tensor.

Relevant Equations

See below.

I derived this equation $$
A_{i,jk}-A_{i,kj}=R^r _{kij}A_r$$.But where do I use this $$A_{i,j}+A_{j,i}=0$$?

 

[Antarres]

In your equation, I assume you use commas to represent covariant derivative(semicolon is usually used for that, while comma indicates a normal partial derivative).

The equation you derived is sometimes used as the definition of Riemann tensor(holds for every vector field). So you're still long way to go from deriving the final identity.

First idea would be to pick the defining equation$A_{i;j} + A_{j;i} = 0$ and act on it with covariant derivative. That way you'd obtain an identity with double covariant derivatives. Relabeling indices circularly, you'll get equivalent identities which you can then sum(or subtract) to obtain more useful identity.

From there on you use the equation you derived, along with symmetries of Riemann tensor to arrive at the final equation. The derivation is not too long, but requires multiple steps which are not very straightforward(or it wasn't for me when I first derived it), but hopefully this hint will set you on the path to get it right.